Question

$If space straight y equals square root of log space straight x plus square root of log space straight x plus square root of log space straight x plus........ infinity end root end root end root comma space prove space that space left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals 1 over straight x$

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Solution

because space straight y equals square root of log space straight x plus square root of log space straight x plus square root of log space straight x plus........ infinity end root end root end root therefore space straight y equals square root of log space straight x plus straight y end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space straight y squared equals log space straight x plus straight y Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get 2 straight y dy over dx equals 1 over straight x plus dy over dx space space space rightwards double arrow left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals 1 over straight x.

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